A vanishing theorem for sheaves of small differential operators in positive characteristic

TL;DR

This paper proves a vanishing theorem for sheaves of small differential operators on certain varieties in positive characteristic, leading to the construction of tilting bundles via Frobenius pushforward.

Contribution

It establishes a new vanishing result for cohomology of small differential operators on specific varieties, enabling the construction of tilting bundles in positive characteristic.

Findings

Vanishing of higher cohomology groups for small differential operators on incidence varieties and quadrics.

Frobenius pushforward of the structure sheaf forms a tilting bundle under certain conditions.

Results depend on the characteristic being larger than the Coxeter number.

Abstract

Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators, and ${\bar D}_X$ its central reduction, the sheaf of small differential operators. In this paper we show that if $X$ is a line-hyperplane incidence variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary dimension (in this case the characteristic is supposed to be odd) then ${\rm H}^{i}(X,{\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived localization theorem for crystalline differential operators (\cite{BMR}) we show that the Frobenius pushforward of the structure sheaf is a tilting bundle on these varieties, provided that $p>h$, the Coxeter number of the corresponding group.